ON THE MOTION OF PENDULUMS. 121 performed on purpose to investigate the decrement of the arc of vibration. It was to be expected beforehand that the results of calculation would fall short of those of observation, inasmuch as only two arcs were registered in each experiment, so that no data were afforded for eliminating the effect of that part of the resistance which did not vary as the first power of the velocity. 78. I have now finished the comparison between theory and experiment, but before concluding this Section I will make a few general remarks. When a new theory is started, it is proper to enquire how far the theory does violence to the notions previously entertained on the subject. The present theory can hardly be called new, because the partial differential equations of motion were given nearly thirty years ago by Navier, and have since been obtained, on different principles, by other mathematicians ; but the application of the theory to actual experiment, except in some doubtful cases relating to the discharge of liquids through capillary tubes, and the determination of the numerical value of the constant p, are, I believe, altogether new. Let us then, in the first instance, examine the magnitude of the tangential pressure which we are obliged by theory to suppose capable of existing in air or water. For the sake of clear ideas, conceive a mass of air or water to be moving in horizontal layers, in such a manner that each layer moves uniformly in a given horizontal direction, while the velocity increases, iii going upwards, at the rate of one inch per second lor each inch of ascent. Then the sliding in the direction of a horizontal plane is equal to unity, and therefore the tangential pressure referred to a unit of surface is equal to u or pp. The absolute magnitude of this unit sliding evidently depends only on the arbitrary unit of time, which is here supposed to be a second. In the case supposed, it will be easily seen that the particles situated at one instant in a vertical line are situated at the expiration of one second in a straight line inclined at an anjjle of 45 to the horizon. Equating the tangential pressure, p'p <„ the normal pressure due to a height h of the ilnid, we ^et k -//>', </ being the force of gravity. Putting now <j = :}<S(>, ^ „ (<)'l !<>)- for air, // = (0'0564)'2 for water, we get //. = 6'000(m,S(> inch for a,ir and h = 0-000008241 inch for water, or about the one thirty-tliou-